If 100 grams of carbon-14 decays until only 25 grams of carbon is left after 11,460 years, what is the half-life of carbon-14?
Start with 100 g so @ 1 half life will be 50 g remaining and 2 half lives will be 25 g remaining; therefore, it has gone through 2 half lives. so
11,460 years divided by 2 = 11,460/2 = half life = ?
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Well, according to my calculations, it seems like carbon-14 has a half-life of being extremely popular among single carbon atoms. It starts off with 100 grams of carbon-14, but after 11,460 years, it's only left with 25 grams of carbon. That's quite the transformation! So, in conclusion, the half-life of carbon-14 is a time period during which it goes from being the life of the party to just another average carbon atom. Talk about peer pressure!
To determine the half-life of carbon-14, we need to understand what half-life means. The half-life of a radioactive substance is the time it takes for half of the initial amount to decay.
In this case, we know that the initial amount of carbon-14 is 100 grams, and after 11,460 years, only 25 grams remain. This means that half of the original 100 grams has decayed.
To find the half-life, we can use the following formula:
Final amount = Initial amount * (1/2)^(time/half-life)
We are given the final amount (25 grams), the initial amount (100 grams), and the time (11,460 years). We can rearrange the formula to solve for the half-life:
(1/2)^(time/half-life) = Final amount / Initial amount
Taking the logarithm of both sides, we have:
(log (1/2)^(time/half-life)) = log (Final amount / Initial amount)
Using the logarithm property log (a^b) = b * log(a), we can simplify further:
(time/half-life) * log (1/2) = log (Final amount / Initial amount)
Now we can isolate the half-life:
half-life = time / (log (Final amount / Initial amount) / log (1/2))
Plugging in the given values:
half-life = 11,460 years / (log (25/100) / log (1/2))
Evaluating the logarithms:
half-life ≈ 11,460 years / (-0.602 / -0.301)
Simplifying further:
half-life ≈ 11,460 years / 2
Finally, we can calculate the half-life:
half-life ≈ 5,730 years
Therefore, the half-life of carbon-14 is approximately 5,730 years.